L(s) = 1 | − 3.46·5-s + 10i·7-s + 13.8i·11-s − 6.92·13-s − 30·17-s + 6.92i·19-s − 12i·23-s − 13.0·25-s − 51.9·29-s − 14i·31-s − 34.6i·35-s + 55.4·37-s − 6·41-s + 62.3i·43-s − 84i·47-s + ⋯ |
L(s) = 1 | − 0.692·5-s + 1.42i·7-s + 1.25i·11-s − 0.532·13-s − 1.76·17-s + 0.364i·19-s − 0.521i·23-s − 0.520·25-s − 1.79·29-s − 0.451i·31-s − 0.989i·35-s + 1.49·37-s − 0.146·41-s + 1.45i·43-s − 1.78i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1382835617\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1382835617\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.46T + 25T^{2} \) |
| 7 | \( 1 - 10iT - 49T^{2} \) |
| 11 | \( 1 - 13.8iT - 121T^{2} \) |
| 13 | \( 1 + 6.92T + 169T^{2} \) |
| 17 | \( 1 + 30T + 289T^{2} \) |
| 19 | \( 1 - 6.92iT - 361T^{2} \) |
| 23 | \( 1 + 12iT - 529T^{2} \) |
| 29 | \( 1 + 51.9T + 841T^{2} \) |
| 31 | \( 1 + 14iT - 961T^{2} \) |
| 37 | \( 1 - 55.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 84iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 17.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 62.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 96.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 48.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 60iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 86T + 5.32e3T^{2} \) |
| 79 | \( 1 + 38iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 13.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 78T + 7.92e3T^{2} \) |
| 97 | \( 1 - 62T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.652116663742431685948823533364, −7.88691221591311851480868318730, −7.14668865259208752533276143248, −6.31992449616549241419187251319, −5.43130742508293413936809193277, −4.58006806654943098508296978420, −3.88251855168438108435987678511, −2.45365696188113506739933752585, −2.04072238375113022808650162438, −0.04365533444592473279095470707,
0.78868126268888265057129419050, 2.24964764112346689184497020241, 3.54512455862346784172102069312, 4.02133597114845622535346211034, 4.89226238519089421566075969861, 5.97562943403546011999093393079, 6.86950168518865883940903314179, 7.47562296393146571029787123377, 8.119958747255047620688117140295, 9.005166353829383169784800389866